
theorem Th27:
  for C be compact non vertical non horizontal Subset of TOP-REAL
2 for n,m be Nat st m >= n & n >= 1 holds X-SpanStart(C,m) = 2|^(m-'
  n)*(X-SpanStart(C,n)-2)+2
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  let n,m be Nat;
A1: X-SpanStart(C,n) = 2|^(n-'1) + 2 by JORDAN1H:def 2;
  assume
A2: m >=n;
  assume
A3: n >= 1;
  then (m-'n)+(n-'1) = (m-'n)+(n-1) by XREAL_1:233
    .= (m-'n)+ n-1
    .= m-1 by A2,XREAL_1:235
    .= m-'1 by A2,A3,XREAL_1:233,XXREAL_0:2;
  then 2|^(m-'1) = 2|^(m-'n)*(X-SpanStart(C,n)-2) by A1,NEWTON:8;
  hence thesis by JORDAN1H:def 2;
end;
